| Property | Level | Notes |
|---|---|---|
| a_b | Species | Intercept of intrinsic growth rate - temperature function of species i. |
| b_{opt,i} | Species | Temperature at which intrinsic growth rate is highest for species i. |
| s_i | Species | Width of intrinsic growth rate - temperature function of species i. |
| a_d | Species | Intercept of death rate - temperature function of species i. |
| z_i | Species | Slope of death rate - temperature function of species i. |
| alpha_{i,j} | Interaction | Strength of interspecific effect of species j on species i. |
Response traits and stability in the context of a pulse perturbation. Are fundamental niche- or realised niche-based response traits more informative?
1 Introduction
In the context of a pulse perturbation we ask:
- will fundamental niche-based response traits, such as those measured according to Ross et al (2023), explain variation in community stability?
- is a realised niche-based response trait able to explain more variation in community stability?
We explore these questions using a simulation model of a multi-species model, in which species intrinsic growth rate is a bell-shaped function of temperature, and each species can have a different temperature optimum. A pulse perturbation is created by briefly setting temperature lower than the control temperature, and then returning it to control temperature.
We then manipulate the strength of inter-specific interactions and examine how that affects the explanatory power of the two types of response trait.
Predictions:
- Realised and fundamental niche-based response traits will provide equal and high explanatory power when there is no or very weak interspecific interactions.
- Increase in strength of interspecific interactions will decrease the explanatory power provided by fundamental niche-based response traits.
- Increase in strength of interspecific interactions will not decrease, or will decrease relatively slowly (compared to fundamental niche-based response traits) the explanatory power provided by realised niche-based response traits.
- Realised niche-based response traits will provide greater explanatory (than fundamental niche-based response traits) when interspecific interactions are relatively strong.
2 Methods
2.1 The multispecies model
We have \(S\) species that can be interacting and whose vital rates are temperature dependent. We assume density-dependent birth rate, (\(B\)), and death rate (\(D\)) in a discrete-time version of the classical Lotka–Volterra model [@de2013predicting; @vasseur2020impact] to get instantaneous growth rate, \(\tilde{r}_{i}(t)\), for species, \(i\), in year \(t\):
\[\begin{equation}\label{eq.r} \tilde{r}_{i}(t) = ln N_{i}(t+1) - ln N_{i}(t) = B(N_{i}(t),N_{j}(t),T(t)) - D(N_{i}(t),N_{j}(t),T(t))(\#eq:r) \end{equation}\]Here, \(N(t)\) represents the biomass at year \(t\), and \(i,j\) are indices for two different species.
The per-capita birth and death rates for \(i^{th}\) species are represented as:
\[\begin{equation}\label{eq.B} B_{i} = b_{0,i}(T)-\beta (N_{i}+\sum_{i \neq j = 1}^{S} \alpha_{ij}N_{j})(\#eq:B) \end{equation}\] \[\begin{equation}\label{eq.D} D_{i} = d_{0,i}(T)-\delta (N_{i}+\sum_{i \neq j = 1}^{S} \alpha_{ij}N_{j})(\#eq:D) \end{equation}\]where, \(\beta\) and \(\delta\) are density-dependent constants, \(\alpha_{i,j}\) is the competition coefficient between species \(i\) and \(j\), and
\[\begin{equation}\label{eq.b0} b_{0,i}(T) = a_{b} e^{-(T-b_{opt,i})^2/s_{i}} (\#eq:b0) \end{equation}\] \[\begin{equation}\label{eq.d0} d_{0,i}(T) = a_{d} e^{z_{i}T} (\#eq:d0) \end{equation}\]with \(a_{b}\), \(a_{d}\) as intercepts, and for \(i^{th}\) species, \(b_{opt,i}\) is the temperature that optimizes birth rate, \(s_{i}\) governs the breadth of the birth function, \(z_{i}\) scales the effect of temperature (in °C) to mimic the Arrhenius relationship.
Substituting Eqs. (\(\ref{eq.B}\)) - (\(\ref{eq.d0}\)) in Eq. (\(\ref{eq.r}\)), we get the following,
\[\begin{equation}\label{eq.r2} \tilde{r}_{i}(t) = r_{m,i} \left( 1-\dfrac{\sum_{i, j = 1}^{S} \alpha_{ij}N_{j}}{K_{i}} \right) (\#eq:r2) \end{equation}\]where, \(\alpha_{ii} = 1\), \(r_{m,i} = (b_{0,i} - d_{0,i})\) is the intrinsic (maximum) rate of increase and \(K_{i} = r_{m,i}/(\beta+\delta)\) is the carrying capacity for the \(i^{th}\) species, respectively.
If abundance drops very low then it is reset to 1. May need to change this, or at least present a good ecological justification.
2.2 Model parameters
The following need to be specified for a community (in addition to the number of species):
2.3 Experimental treatments
2.3.1 Pulse pertubation treatment
The context of our entire study is a pulse perturbation. Temperature is constant in one treatment, and is pulse perturbed in another. For simplicity, we always make the perturbed temperature lower than the control temperature. We do not manipulate or change anything about the pulse perturbation. The graph below shows the control temperature in black and the pulse in red. Simulations are run for 10’000 time steps before the pulse is applied and run for another 1’000 after the pulse starts.
2.3.2 Creating a community
In order to create a community we define a set of \(S\) species and in the following only two things differ among the species:
- The optimum temperature for growth, such that some species have maximum intrinsic growth rate at higher temperatures and some have maximum intrinsic growth at lower temperatures. The optimum temperature for growth for each species in a community is drawn from a uniform distribution with a specific mean and range.
- The interspecific interactions in the interaction matrix \(alpha_{i,j}\). These are drawn from a normal distribution with mean zero and standard deviation \(sd(alpha_{i,j})\).
2.3.3 Community composition treatment
We create communities that vary in their composition by changing the mean of the uniform distribution from which species’ optimum temperature for growth are drawn. If we set the mean low, close to the temperature of the perturbation, then the perturbation as a positive effect on many of the species intrinsic growth rates. In contrast, if we set the mean close to the control temperature then the perturbation has mostly negative effects on the species’ intrinsic growth rates. We show some communities in the Results section of this report.
2.3.4 Strength of species interactions treatment
When we set \(sd(alpha_{i,j})\) = 0 then there are no interspecific interactions. Larger values of \(sd(alpha_{i,j})\) mean that interspecific interactions are stronger.
2.4 Analyses
2.4.1 Species response traits
We calculated two types of species response trait. First was a response trait based on the temperature response curve of a species (i.e., as in Ross et al (2023)). For this trait, we calculated the effect of the perturbation on intrinsic growth rate (perturbation temperature intrinsic growth rate - control temperature intrinsic growth rate). This is analogous to the calculating the first derivative (slope) of the temperature response curve.
Second was a response trait based on the observed response of the species to the pulse perturbation in the community context. Hence this response trait depends on the direct effect of the pulse on a species, and also the indirect effects via other species in the community.
2.4.2 Community response diversity
Currently we calculate:
- IGR-effect trait mean, variance, dissimilarity response diversity, and divergence response diversity.
- Species AUC trait mean and variance.
2.4.3 Community stability
This was calculated as the effect of the pulse perturbation on the total community biomass. This effect varied through time (e.g., often decreased monotonically after the pulse ended). The effect was calculated as the control - perturbed abundance and could conceivably be sometimes positive and sometimes negative. Examples of the dynamics of total biomass in control and perturbed treatments are given in the Appendix. A schematic figure is just below.
In one measure of community stability, we took the sum (over time) of the effect. If the effect was always positive, the summed effect would be positive and if the effect was always negative then the summed effect would be negative. A small (close to zero) summed effect could result from a balance of positive and negative effects, or just very small but consistent positive (or negative) effects. Stability is greatest when the summed effect is zero. More negative and more positive values imply lower stability. We call this raw community stability.
Another measure was the sum of the absolute effect size (OEV). This will only be small (closer to zero) when the perturbation has a consistently small effect (positive, or negative, or mix of positive and negative). A mix of large positive and large positive effect would lead to a large value of this measure. We call this absolute community stability.
3 Results
3.1 Explanatory power
And here we show how much variability in community stability is explained by the community mean of the species response traits, for realised niche-based response traits, and for fundamental niche-based response traits. How much variability explained is that of a GAM.
Seems surprising that the fundamental niche-based response traits always (on average) provide greater explanatory power. This contradicts expectations in the Introduction section (Section 1).
3.2 Response trait and community stability
In the following various graphs is the relationship between community mean of species response trait (realised or fundamental) (x-axis) and community stability (y-axis).
Each panel is for a different strength of interspecific interaction.
3.3 Some individual communities
Examples 1, 2, and 3 are with no interspecific interactions. The first and second are with relatively low variation in species responses, whereare the third has higher variation (half positive and half negative). Notice how the resistance of total biomass of the third community is greater than the resistance of the first two.
Examples 4 has relatively weak interspecific interaction, and example 5 has relatively strong interspecific interaction. The black boxes in some of the individual species responses are the addition of 1 abundance unit when the abundance drop too low (see last sentence in section Section 2.1).
3.3.1 Example 1
No inter-specific interactions (sd(alpha_ij) = 0) and low average optimum temperature (mean(b_opt) = 17), so the pulse has positive effect on intrinsic growth rate of most species.
| species_id | species_tot_deltabm_spline | species_tot_deltabm_raw | igr_pert_effect |
|---|---|---|---|
| Spp1 | 397.323126 | 38.897582 | 2.6299985 |
| Spp2 | 92.905271 | 9.145326 | 1.4892266 |
| Spp3 | 236.847663 | 23.258841 | 2.2206364 |
| Spp4 | 726.783662 | 70.627543 | 2.9749665 |
| Spp5 | 700.982850 | 68.161700 | 2.9676157 |
| Spp6 | 397.413906 | 38.906397 | 2.6301713 |
| Spp7 | -15.319305 | -1.510352 | -0.9896932 |
| Spp8 | 453.211181 | 44.317104 | 2.7270513 |
| Spp9 | 482.600726 | 47.161108 | 2.7711554 |
| Spp10 | 7.901579 | 0.778779 | 0.2710188 |
3.3.2 Example 2
No inter-specific interactions (sd(alpha_ij) = 0) and low average optimum temperature (mean(b_opt) = 21), so the pulse has negative effect on intrinsic growth rate of most species.
| species_id | species_tot_deltabm_spline | species_tot_deltabm_raw | igr_pert_effect |
|---|---|---|---|
| Spp1 | -10.66239 | -1.051154 | -0.5664323 |
| Spp2 | -21.94933 | -2.164210 | -2.5705698 |
| Spp3 | -22.70787 | -2.239026 | -2.7299823 |
| Spp4 | -21.94583 | -2.163865 | -2.5684242 |
| Spp5 | -22.28319 | -2.197139 | -2.7953846 |
| Spp6 | -14.20979 | -1.400942 | -0.8701080 |
| Spp7 | -22.51443 | -2.219946 | -2.9542994 |
| Spp8 | -22.09460 | -2.178539 | -2.6635800 |
| Spp9 | -22.72304 | -2.240522 | -2.5614904 |
| Spp10 | -22.71672 | -2.239898 | -2.6445156 |
3.3.3 Example 3
No inter-specific interactions (sd(alpha_ij) = 0) and intermediate average optimum temperature (mean(b_opt) = 19), so the pulse has negative effect on intrinsic growth rate of some species, and positive effect on others.
| species_id | species_tot_deltabm_spline | species_tot_deltabm_raw | igr_pert_effect |
|---|---|---|---|
| Spp1 | -21.8809418 | -2.1574650 | -2.5294106 |
| Spp2 | -0.2768503 | -0.0272894 | -0.0110880 |
| Spp3 | -10.6198777 | -1.0469620 | -0.5633309 |
| Spp4 | 148.5968090 | 14.6144606 | 1.8478451 |
| Spp5 | 37.0129725 | 3.6464880 | 0.8828283 |
| Spp6 | 160.6439637 | 15.7961656 | 1.9094060 |
| Spp7 | -22.3788858 | -2.2065772 | -2.8650254 |
| Spp8 | 13.8430049 | 1.3642519 | 0.4328663 |
| Spp9 | 49.0212389 | 4.8286894 | 1.0511240 |
| Spp10 | -21.7258368 | -2.1421672 | -2.4418049 |
3.3.4 Example 4
“Weak” inter-specific interactions (sd(alpha_ij) = 0.1) and intermediate average optimum temperature (mean(b_opt) = 19), so the pulse has negative effect on intrinsic growth rate of some species, and positive effect on others.
| species_id | species_tot_deltabm_spline | species_tot_deltabm_raw | igr_pert_effect |
|---|---|---|---|
| Spp1 | 138.771851 | 13.6694244 | 1.6868698 |
| Spp2 | -8.793688 | -0.8850898 | 0.1001117 |
| Spp3 | -20.867889 | -2.0626095 | -2.0523381 |
| Spp4 | -18.735549 | -1.8460012 | -1.9941551 |
| Spp5 | 15.180029 | 1.4937418 | 0.6072451 |
| Spp6 | -16.879411 | -4.5755863 | -0.2140441 |
| Spp7 | -25.989264 | -2.5547891 | -2.8195527 |
| Spp8 | 909.728664 | 87.4912990 | 1.0236420 |
| Spp9 | -25.938320 | -2.5540767 | -1.6919063 |
| Spp10 | -21.382427 | -2.1177404 | -2.1266226 |
3.3.5 Example 5
“Strong* inter-specific interactions (sd(alpha_ij) = 0.3) and intermediate average optimum temperature (mean(b_opt) = 19), so the pulse has negative effect on intrinsic growth rate of some species, and positive effect on others.
| species_id | species_tot_deltabm_spline | species_tot_deltabm_raw | igr_pert_effect |
|---|---|---|---|
| Spp1 | -47.084571 | -4.7699092 | 0.4429142 |
| Spp2 | 237.120427 | 23.3848559 | 2.0794836 |
| Spp3 | 19.405517 | 1.9391437 | 0.2233202 |
| Spp4 | -2.608923 | -0.2409980 | -1.0666445 |
| Spp5 | -48.132551 | -4.7638183 | -2.0736713 |
| Spp6 | -20.698595 | -2.0429755 | -2.2457595 |
| Spp7 | -20.893100 | -2.7979771 | -0.2194850 |
| Spp8 | 27.049410 | 0.0332636 | -0.2054862 |
| Spp9 | 15.805815 | 1.5536970 | 1.3014851 |
| Spp10 | 59.186443 | 3.7161193 | 1.6706320 |
4 Appendix
4.1 Check various relationships
